Acta Biotheor (2011) 59:53–66
DOI 10.1007/s10441-010-9107-8
REGULAR ARTICLE
The Sunk-cost Effect as an Optimal Rate-maximizing
Behavior
Theodore P. Pavlic • Kevin M. Passino
Received: 7 March 2009 / Accepted: 30 June 2010 / Published online: 24 July 2010
Springer Science+Business Media B.V. 2010
Abstract Optimal foraging theory has been criticized for underestimating patch
exploitation time. However, proper modeling of costs not only answers these criticisms, but it also explains apparently irrational behaviors like the sunk-cost effect.
When a forager is sure to experience high initial costs repeatedly, the forager should
devote more time to exploitation than searching in order to minimize the accumulation of said costs. Thus, increased recognition or reconnaissance costs lead to
increased exploitation times in order to reduce the frequency of future costs, and this
result can be used to explain paradoxical human preference for higher costs. In fact,
this result also provides an explanation for how continuing a very costly task
indefinitely provides the optimal long-term rate of gain; the entry cost of each new
task is so great that the forager avoids ever returning to search. In general, apparently irrational decisions may be optimal when considering the lifetime of a forager
within a larger system.
Keywords Solitary animal behavior Patch residence time Rationality Concorde fallacy Escalation error Optimal foraging theory
T. P. Pavlic (&) K. M. Passino
Department of Electrical and Computer Engineering, The Ohio State University, 205 Dreese Labs,
2015 Neil Avenue, Columbus, OH 43210, USA
e-mail: [email protected]
K. M. Passino
e-mail: [email protected]
K. M. Passino
Department of Evolution, Evology, and Organismal Biology, The Ohio State University,
318 W. 12th Avenue, Columbus, OH 43210, USA
123
54
T. P. Pavlic, K. M. Passino
1 Introduction
Influential work by Schoener (1971) postulates that natural selection favors
behaviors on a continuum from foraging time minimization to net energetic gain
maximization. Techniques developed by Pyke et al. (1977) and Charnov (1973) use
gain-to-time rate maximization to quantitatively predict which behavior from this
continuum will be favored in particular cases. These techniques have been
popularized by Stephens and Krebs (1986) as the prey and patch models of classical
optimal foraging theory (OFT). They respectively describe which prey foragers
should include in their diet and how long foragers should exploit a patch of prey,
which are the two central questions of solitary foraging theory.
The rate-maximizing prey and patch models have mixed success in explaining
behavioral observations in the field. The prey model accurately describes patterns
of preference, and the patch model makes accurate predictions about how foraging
durations should change when background parameters change, but the patch model
has poor success in many cases when predicting actual foraging durations (Stephens
and Krebs 1986; Nonacs 2001; Sih and Christensen 2001). In a review by Nonacs
(2001), several examples are picked that show foraging durations tend to be longer
than expected by the classical patch model. Nonacs concludes that OFT is
incomplete and that optimal behavior is not described by rate maximization.
However, recent studies show how observed behaviors that are inconsistent with
classical OFT are indeed rate maximizing under an adjusted foraging model. For
example, shorebirds that appear to violate classical OFT are shown to be rate
maximizers when explicitly modeling digestive constraints (van Gils et al. 2005)
and the value of information discovery (van Gils et al. 2003).
Here, we show how explicitly modeling foraging costs can answer Nonacs’
criticisms of the classical OFT’s underpredictions of foraging duration.
Additionally, we show how the same modifications lead to foraging theoretic
explanations of the sunk-cost effect (Arkes and Blumer 1985; Arkes and Ayton
1999; Kanodia et al. 1989; Staw 1981), which describes behaviors that invest
more time in the more costly of two otherwise equivalent resources. This effect is
also known as the Concorde fallacy (Dawkins and Carlisle 1976) because it
describes an apparent logical fallacy analogous to the continued development of a
supersonic jet that never returns a profit. However, by giving a foraging theoretic
explanation, we show that the fallacy is actually an optimal behavior. This
explanation is also consistent with observations of animals that commit longer
feeding times after moving into areas where prey requires greater energy to
acquire (Nolet et al. 2001). Additionally, if a forager’s uncertainty in estimating
patch quality is modeled as an initial average cost associated with each patch
type, the extended foraging durations predicted by our augmented OFT is
consistent with the ‘‘wait for good news’’ behaviors predicted and observed in
supposed Bayesian foragers (Olsson and Holmgren 1998; van Gils et al. 2003;
Olsson and Brown 2006).
This paper is organized as follows. First, in Sect. 2, we summarize the principal
results from classical OFT. Next, in Sect. 3, we present a common empirical
criticism of OFT and provide a mathematical response supporting OFT. In Sect. 4,
123
The Sunk-Cost Effect as an Optimal Rate-Maximizing Behavior
55
we augment the traditional graphical analysis method from OFT to include effects
of search and in-patch costs. This analysis shows how explicitly including costs
leads to longer patch residence times that are more consistent with observed
behavior. We use similar graphical methods in Sect. 5 to provide a detailed
explanation of the sunk-cost effect as an adaptive rate-maximizing behavior.
Finally, in Sect. 6, we give some concluding remarks.
2 Classical Optimal Foraging Theory
Optimal foraging theory models a forager that faces n 2 N types of prey. The
forager encounters prey of type i [ {1, 2, …, n} according to an independent
Poisson counting process with rate ki [ 0. Each encountered prey item of type i is
picked for handling with probability pi [ [0,1], and those chosen items are
exploited for si C 0 time on average. The forager pays a cost (e.g., energy) of cs per
unit search time, but it receives an average reward of gi(si) after handling an item of
type i. Assuming that natural selection favors foragers that maximize their lifetime
gain, behaviors should trade increased reward per encounter for increased number of
encounters in a lifetime. So the optimal behavior will maximize the rate
P
cs þ ni¼1 ki pi gi ðsi Þ
P
:
ð1Þ
1 þ ni¼1 ki pi si
The prey model treats the case where si is fixed for each type i and the forager
must make a binary choice to handle or ignore each encountered prey. The patch
model treats the opposite case where p1 = p2 = _ = pn = 1 (i.e., all encountered
prey are exploited) and the forager must choose when to stop exploiting each prey.
A central result of the patch model is the marginal value theorem (MVT) (Charnov
1973, 1976), which states that nonzero exploitation times s1 ; s2 ; . . .; sn that make
P
cs þ ni¼1 ki gi ðsi Þ
0 P
gi ðsi Þ ¼
and g00i ðsi Þ\0
ð2Þ
1 þ ni¼1 ki si
for each type i will correspond to an optimal patch foraging behavior. That is,
behaviors that equate per-prey rate of gain with per-lifetime rate of gain will
appropriately balance present reward with future opportunity cost so as to be
optimal over a lifetime.
3 OFT Criticism and Explicit Processing Costs
Nonacs (2001) bases his criticisms on the observation that Eq. 2 implies that
g01 ðs1 Þ ¼ g02 ðs2 Þ ¼ ¼ g0n ðs Þ: That is, if behaviors in nature could be described
by the MVT, then every per-type rate of average gain (i.e., ‘‘speed’’ of gain) should
be equal to some global giving-up density (GUD). In the studies that Nonacs
reviews, foragers stop exploiting different prey types at different speeds, which
leads him to the conclusion that the MVT does not hold in reality.
123
56
T. P. Pavlic, K. M. Passino
As discussed by Stephens and Krebs (1986) the function gi is not meant to be a
gross observable reward to the forager. Instead, it models the energetic reward to
the forager minus the internal handling cost, which is usually not externally
observable. To make this distinction clearer, we explicitly introduce an average
handling cost ci(si) for exploiting prey of type i for an average of si time, and we
let gi(si) be the corresponding average observable reward. Thus, the forager should
maximize
P
cs þ ni¼1 ki pi ðgi ðsi Þ ci ðsi ÞÞ
P
:
ð3Þ
1 þ ni¼1 ki pi si
In this framework, the MVT states that times s1 ; . . .; sn that make
P
cs þ ni¼1 ki gi ðsi Þ ci ðsi Þ
0 0 P
gi ðsi Þ ci ðsi Þ ¼
and g00i ðsi Þ c00i ðsi Þ\0 ð4Þ
1 þ ni¼1 ki si
will correspond to an optimal patch behavior. From Eq. 4, it must be that g01 ðs1 Þ c01 ðs1 Þ ¼ g02 ðs2 Þ c02 ðs2 Þ ¼ ¼ g0n ðsn Þ c0n ðs2 Þ: The rare case when the observable rewards are such that g01 ðs1 Þ ¼ g02 ðs2 Þ ¼ ¼ g0n ðsn Þ is when the internal
handling cost rates are also all equal, and so a global GUD is exceptional.
Differences in terminal handling densities across types reflect differences in the
handling burden of those types. For example, for types with the same reward–time
curve and linear handling cost functions (i.e., ci ðsi Þ,ci si for type i and constant
ci [ 0), smaller marginal handling costs (i.e., c0i ðsi Þ ¼ ci for type i) should lead to
longer optimal exploitation times—foragers spend relatively less time in patches
with steep handling-cost functions.
4 Graphical Optimization and Long Residence Times
The intuitive graphical analysis approach frequently used by Stephens and Krebs
(1986) for the single-type case can be extended to our model, which makes all costs
explicit and uses multiple types. To do so, we define k,k1 þ k2 þ þ kn to be the
average Poisson encounter rate for all prey types combined. Then 1/k is the average
search time between encounters, and
n
n
n
X
X
X
ki
ki
ki
pi gi ðsi Þ; c,
pi ci ðsi Þ; and s,
pi s i
g,
k
k
k
i¼1
i¼1
i¼1
are the average per-encounter handling gain, cost, and time. Equation 3 then becomes
s
g c ck
:
1
kþs
ð5Þ
Graphical optimization of Eq. 5 is shown in Fig. 1.
Each point in the shaded area corresponds to a particular choice of preferences,
p1, p2, …, pn, and exploitation times, s1,…, sn. The upper boundary of the shaded
area is the optimal frontier on which all optimal behaviors are found. Optimization
consists of rotating a ray originating from (-1/k,cs/k) counter-clockwise from a
123
The Sunk-Cost Effect as an Optimal Rate-Maximizing Behavior
57
Fig. 1 Effect of searching on
optimal patch residence times.
As search cost cs or average
interarrival time 1/k increases,
the slope R* of the optimal ray
will decrease, and the optimal
average exploitation time s will
increase
-90 angle to the least upper bound of the angles less than 90 where the ray
intersects the shaded region. The slope of the optimal ray R* is the optimal value of
Eq. 5, and the corresponding values ðg cÞ and s are the optimal per-encounter
average net handling gain and exploitation time. Hence, this graphical interpretation
of rate maximization allows for predictions of how perturbing parameters of the
model will impact the average handling time s: An increase in the average handling
time may be due to an increase in the optimal exploitation time for each patch (i.e.,
changes in si for each type i) or due to additional patch types being added to the
optimal diet (i.e., changes in pi for each type i) or a mixture of the two effects. This
graphical approach does not suggest how to predict each individual si or pi ; it is only
meant to investigate how certain parameter perturbations will affect general foraging
trends. Specific optimal diet content and patch exploitation times can be found using
algorithms from classical OFT (Charnov 1973, 1976; Stephens and Krebs 1986;
Pavlic 2007).
When Stephens and Krebs (1986) consider graphical solutions to the single-type
case, they frequently discuss how changes in encounter rate k1 lead to changes in
optimal behavior. For multiple types, the overall rate k can be changed while holding
the density ki/k constant for each type i. The graphical example in Fig. 1 shows that
the optimal average exploitation time s will increase if either the overall encounter
rate k decreases or the search cost rate cs increases. In both cases, to maximize lifetime
reward, the forager must increase present gains to compensate for future search
losses—the opportunity cost of more exploitation decreases when search time or cost
increase. Likewise, if the average per-encounter handling cost c is increased, the g c
curve will be shifted down the vertical axis and the optimal average exploitation time
s will increase. This prediction suggests an explanation for the ostensibly anomalous
observation by Nolet et al. (2001) that tundra swans spend more time feeding in areas
of deep water where more energy is required to acquire similar prey as in shallow
water. The increased costs necessarily decrease the maximal long-term rate of gain,
which is the opportunity cost of increased exploitation, and so patches are exploited
longer. This effect is investigated in more detail in Sect. 5.
5 The Sunk-Cost Effect
Graphical optimization of Eq. 3 helps explain apparently irrational behaviors
discussed in the fields of economics, psychology, and biology in terms of lifetime
123
58
T. P. Pavlic, K. M. Passino
gain maximization. In economics, the propensity to continue a costly task after
paying some initial cost is sometimes called an escalation commitment, escalation
behavior, or escalation error, but it is more commonly known as the sunk-cost effect
(Kanodia et al. 1989; Staw 1981). This nomenclature is also used by psychologists
(e.g., Arkes and Blumer 1985). In fact, psychologists Arkes and Ayton (1999) note
that the sunk-cost effect is equivalent to the Concorde fallacy first described by
biologists Dawkins and Carlisle (1976). Although none of these terms are used, the
same phenomena is also observed by Nolet et al. (2001). In particular, tundra swans
must expend more energy to ‘‘up-end’’ to feed on deep-water tuber patches than
they do to ‘‘head-dip’’ to feed on shallow-water patches; however, contrary to the
expectations of Nolet et al., the swans feed for a longer time on each high-cost
deep-water patch. In every context, the observation of the sunk-cost effect is an
enigma because intuition suggests that this behavior is suboptimal. Here, we show
how optimization of Eq. 3 predicts the sunk-cost effect for certain scenarios; a
common element of every case is a large initial cost.
5.1 Initial Costs: Recognition, Acquisition, Reconnaissance
For simplicity, we make a graphical argument with the assumptions that n = 1,
cs = 0, and p1 = 1 (i.e., the patch case). We also revert to interpreting g1 as the
average net handling gain (i.e., the sum of an observable gain and an internal cost).
Under these assumptions,
g1 ðs1 Þ
1
k 1 þ s1
ð6Þ
is the objective function for optimization. Similar arguments can be made about the
general multivariate form in Eq. 3.
Consider the case when gain function g1 is initially:
(1)
(2)
(3)
negative (i.e., g1(0) \ 0),
increasing (i.e., g01 ð0Þ [ 0), and
decelerating (i.e., g001 ð0Þ\0Þ
Functions meeting items (2) and (3) are treated by Stephens and Krebs (1986).
However, these functions are all initially zero. Here, we let the average handling
gain g1(s1) be initially negative to reflect some initial cost. For example, rather than
treating recognition cost as a separate quantity (e.g., Stephens and Krebs 1986,
pp. 79–81), we consider them to be a characteristic of the gain returned to the
forager from handling each item (e.g., an initial energy expenditure required simply
for access to or identification of a prey). Alternatively, as in the case of the tundra
swans observed by Nolet et al. (2001), the initial cost may model the extra energy
required to acquire prey (e.g., by ‘‘up-ending’’ instead of simply ‘‘head-dipping’’).
Furthermore, just as Olsson and Brown (2006) show how Bayesian foragers receive
a relative ‘‘foraging benefit of information’’ for spending extra time in a patch, the
initial cost here may be viewed as an average penalty due to uncertainty about the
patch gain function.
123
The Sunk-Cost Effect as an Optimal Rate-Maximizing Behavior
59
Fig. 2 Different recognition
costs r2 [ r1 [ r0 have different
optimal exploitation times
s12 [ s11 [ s10 and optimal
rates R2 \R1 \R0 : For two
equally shaped gain functions,
the one with the higher initial
cost will also have the higher
optimal residence time
(a)
(b)
(c)
The graphical optimization procedure for such a initially negative gain function
is shown in Fig. 2. The same gain function is shown in Figs. 2a–c except that initial
recognition cost is low, moderate, and high, respectively. When the initial cost
increases from r0 to r1, the optimal exploitation time increases from s10 to s11 : Even
when the initial cost increases from r1 to r2 and makes the gain function strictly
negative, the exploitation time still increases from s11 to s12 : This propensity to
exploit items longer when the initial cost is increased is exactly the sunk-cost effect
(Arkes and Ayton 1999; Arkes and Blumer 1985). However, it also maximizes the
long-term net rate of gain, and so it is the rational behavior.
This result can be explained using an opportunity cost interpretation (Houston
and McNamara 1999; Stephens and Krebs 1986). The optimal rate R* is an
opportunity cost for spending additional time exploiting an item. It represents the
maximum expected gain possible per unit time, and so it is costly to exploit items of
a particular type for so long that the type’s rate of average gain falls below R*. When
the initial cost of handling increases on all encounters, the total gain decreases, and
so the opportunity cost for additional exploitation decreases (e.g., R1 \R0 ). Thus,
the marginal handling gain must decrease further in order for marginal costs and
123
60
T. P. Pavlic, K. M. Passino
marginal benefits to equalize. This effect manifests itself through the increase in
optimal exploitation time.
By increasing exploitation time when initial cost increases, the forager reduces
the amount of time it spends paying high costs. Leaving patches at an earlier time is
equivalent to volunteering for paying recognition costs more frequently. It is not the
increased initial cost on a single encounter that is important; it is the increased initial
cost on all encounters that causes the decrease in opportunity cost for additional
exploitation.
5.2 Human and Nonhuman Examples
5.2.1 Feeding Time Increases in Areas of High-Cost Prey
In a study by Nolet et al. (2001), the foraging behavior of tundra swans in shallow
water is compared to the behavior in deep water. The swans forage on belowground
tubers, and thus the swans need only to ‘‘head-dip’’ in shallow water whereas they
need to ‘‘up-end’’ in deep water to find and retrieve the tubers. The model that the
authors use to explain the tundra swan behavior predicts that there will be a decrease
in feeding time in areas where feeding has larger power requirements. However, in
deep water where the power requirements to up-end are apparently larger than the
requirements to head-dip, the observed tundra swans spent longer times feeding on
tuber patches. Although this result is a contradiction of the model used by Nolet
et al., it follows directly from the discussion in Sect. 5.1. In particular, let n C 1
(i.e., multiple tuber patch types are available) and p1 = _ = pn = 1 (i.e.,
encounters are not ignored). For each patch type i, the average net gain function
gi is assumed to be initially negative in order to model the energetic burden of
acquiring the belowground tubers. Hence, we use the classical marginal value
theorem result in Eq. 2 applied to initially negative gain functions.
The curve in Fig. 3a models the average value gi of a tuber patch of type i after
exploiting the patch for time si. The average cost ci of a patch entry reduces the
maximum possible long-term rate of gain R* from all patches. The MVT predicts
that the optimal exploitation time si occurs when g0i ðsi Þ ¼ R ; which is depicted in
Fig. 3b for three different R* maximum rates corresponding to three different ci
entry costs. As the average entry cost of a patch type increases, the maximum rate
decreases, and so average exploitation time increases. When R* is very high either
due to low cost ci or high return rates from other patches, the optimal commitment
time drops to zero because the shallow value function is dominated by steep returns
from other options.
5.2.2 Human Preference for High Price
One recorded example of the sunk-cost effect in humans comes from studying
behavior at the cinema. In a controlled study, Arkes and Blumer (1985) charge
movie patrons different prices for tickets to see a particular movie. They show that
the likelihood that people attend the movie is positively correlated with the cost of
the ticket, and they conclude that this behavior is an irrational mistake. However,
123
The Sunk-Cost Effect as an Optimal Rate-Maximizing Behavior
61
(a)
(b)
Fig. 3 Optimal exploitation time for patch type i with acquisition cost ci. Increased costs lead to
decreased overall return rate R*, which leads to greater exploitation time in each costly patch. That is,
when most encountered patches are costly to enter, exploitation time in each of these high-cost patches
will be long in order to reduce their frequency. However, when enough other low-cost patches are
available to raise the overall return rate sufficiently high, exploitation time to patches with relatively
shallow marginal returns may decrease to zero. That is, when there are enough high-quality patches, time
is better spent searching for them
this result can also be predicted by the classical marginal value theorem result in Eq.
(2) applied to initially negative gain functions.1 Again, consider the case where
n C 1 (i.e., multiple types are available) and p1 = _ = pn = 1 (i.e., encounters are
not ignored). For each activity type i, let the gain function gi(si) be parameterized by
commitment time si that represents the average time the individual is committed to
participating in that type of activity. That is, if the commitment time for a type of
activity is low, there is a low probability that the individual will complete the
activity, and thus the average gain returned from this type of activity will also be
low. Likewise, if the commitment time for the activity is high, it will likely be
attended, and so the value returned will level off as it returns no value after it is
attended. Hence, the analysis summarized in Fig. 3 also applies here. As ticket price
1
Here, to be consistent with Arkes and Blumer (1985), we do not allow encounters to be ignored, and so
initial costs are forced and the pure patch model predicts the optimal behavior. The combined prey–patch
model better fits reality as ticket purchasing opportunities can be ignored.
123
62
T. P. Pavlic, K. M. Passino
increases for this set of individuals who are forced to buy the ticket, they become
more strongly committed to attending the movie.2
This example matches classical economic intuition. As the price a consumer pays
increases, the overall purchasing frequency decreases. The longer commitment to a
higher-priced movie reflects reluctance to make more purchases. Over the long
term, this behavior maximizes the net value the consumer holds collectively in
goods and capital. Behaving in any other way would be reckless when viewed from
a long-term opportunity-cost perspective. In particular, exploitation times are longer
in order to accumulate more gain to justify returning to a search that will likely
result in finding more ‘‘prey’’ that force high initial costs. Thus, the sunk-cost effect
may seem nonsensical for any single purchase, but it is adaptive when considering
an economic or ecological system view.
5.3 Escalation Behavior
The propensity to continue a costly task ad infinitum is an extreme form of the sunkcost effect known as escalation behavior or escalation error (Kanodia et al. 1989;
Staw 1981). In fact, the Concorde fallacy gets its name from the example given by
Dawkins and Carlisle (1976) of continued government investment on a supersonic
jet that has no promise of ever returning a profit. Here, we show how escalation
behavior can be an adaptive rate-maximizing strategy under special circumstances.
These examples presume that the optimal long-term rate of gain R* is negative, and
so they are a better fit for economic examples where prolonged deficits are allowed.
However, as several studies show that foragers can dynamically adjust behavior to
maximize the present estimate of long-term rate of gain (Giraldeau and Livoreil
1998; Giraldeau and Caraco 2000; Sih and Christensen 2001), these examples may
also be consistent with short-term trends in behavior when foragers are temporarily
subjected to high-cost environments. Additionally, this analysis may apply to
similar mathematical models of value maximization when behaviors must choose
from a set of costly options. For example, in spiders that choose to join existing
webs or spin their own (Jakob 2004), small spiders face potentially dangerous
competition with large spiders in new webs but also face high costs in building new
webs. Also, studies of women in abusive relationships suggests their decisions to
leave the relationship are affected by fear of violence toward pets that remain (Faver
and Strand 2003). In both of these cases, the optimal behavior may be accompanied
by costs that might seem prohibitive in isolation. As in Sect. 5.1, we use the
simplified optimization objective given by Eq. 6.
In Fig. 4, a positive constant gain function is shown as the limit of asymptotic
gain functions of increasing steepness. Because of the asymptotic upper limit, sharp
increases in gain correspond to sharp decreases in marginal returns. So the optimal
patch residence time decreases as steepness increases because marginal returns fall
so quickly. In the limiting case, when the gain function is constant, the optimal
behavior is to leave immediately because staying longer cannot result in any
2
If the experimenters allowed for encounters to be ignored (i.e., if participants could choose to not
purchase a ticket), movies with zero commitment times would also have zero ticket sales.
123
The Sunk-Cost Effect as an Optimal Rate-Maximizing Behavior
63
Fig. 4 Analysis of positive constant gain function. The gain function is the limit of smooth asymptotic
gain functions of increasing steepness. The limiting optimal behavior is to leave the patch immediately.
Staying longer returns no additional gain, and leaving immediately decreases the time between
subsequent high-quality patches
additional gain. In particular, it is better to invest time searching for new patches
that guarantee an entry reward g at regular rate k1 (i.e., long-term rate of gain k1g)
rather than staying in any single patch that provides no additional gain (i.e., longterm rate of gain g/?& 0).
The opposite effect is demonstrated in Fig. 5, which shows a negative gain
function as the limit of asymptotic gain functions of decreasing steepness. As the
steepness of each gain function is reduced, the optimal patch residence time
increases without bound. So the optimal behavior for a constant gain function that is
negative is to remain forever in each encountered patch. It is better to pay a single
recognition cost once and remain in the patch forever rather than paying the
recognition cost repeatedly at each new encounter.
As shown in the lower portion of Fig. 5, even when the gain function is initially
decreasing and convex, the optimal behavior is not to immediately leave the patch
unless the initial slope is sufficiently negative. It is better for the forager to pay
additional costs within the patch rather than face the -k1g lifetime rate of gain
Fig. 5 Analysis of negative constant gain function. Escalation error can be viewed as the limit
of increasingly shallow gain functions that are initially negative. Zero patch residence time is not restored
until steepness is sufficiently negative. For this class of functions, optimal points will never be found in
the shaded region between the horizontal axis and the constant gain function
123
64
T. P. Pavlic, K. M. Passino
Fig. 6 Initially negative concave gain function has latent escalation behavior that dominates the early
local maximum. Leaving the patch immediately returns a higher long-term gain than staying a short time
longer, but it returns a lower gain than staying for any time greater than T. Hence, escalation behavior is
optimal because it prevents paying any future entry costs
associated with immediately leaving the patch. However, escalation behavior is
avoided in this case because the convex gain function eventually becomes too steep.
A different effect occurs for initially negative and initially steep concave functions
like the one shown in Fig. 6. Here, immediate withdrawal from the patch is a locally
optimal behavior. That is, the objective function decreases as the exploitation time
is increased slightly. However, as exploitation time increases, the objective function
reaches its global minimum and then starts increasing. Moreover, there exists a time
T such that exploiting these patches for any time s1 C T will be favorable to
immediate withdrawal. Again, it is better to exploit patches with a shallow negative
gain longer than to search for more patches that have a steep gain.
In the supersonic jet construction example (Dawkins and Carlisle 1976), the
costly behavior may be adaptive because it prevents starting other costlier ventures.
More generally, successful politicians in power may induce endless and massive
spending in one area for no other reason than to avoid new spending in other areas.
In a foraging context, a forager that must pay a large energetic cost to enter a patch
(e.g., due to high predation risk, having to swim upstream, or high uncertainty about
patch quality) may hesitate to ever leave that patch if the habitat contains few other
types of patches. Similarly, social group members may resist the temptation to leave
a low-quality group if joining any other group is accompanied by costly antagonistic
violence (e.g., Faver and Strand 2003; Jakob 2004). In general, in a set of apparently
bad choices, the choice that has the lowest eventual marginal losses will minimize
the very long-term losses.
6 Conclusions
Optimal foraging theory introduces quantitative analysis into the study of behavior
in a way that complements intuition. We have shown how explicitly modeling
foraging costs can improve the accuracy of rate-maximizing OFT methods and
answer the criticisms of Nonacs (2001) that rate maximization too often
underestimates patch residence time. As verified by van Gils et al. (2003), the
Bayesian forager modeled by Olsson and Holmgren (1998) accurately predicts
observed patch residence times. This result is investigated by Olsson and Brown
(2006) who derive a ‘‘foraging benefit of information’’ that is an additional reward
123
The Sunk-Cost Effect as an Optimal Rate-Maximizing Behavior
65
that Bayesian foragers receive for staying longer in patches to gather information.
As these models investigate foraging behavior using numerical dynamic programming techniques, they lack the intuition of classical OFT. However, if the discovery
penalty of the information gathering process is modeled as an initial in-patch cost,
then relatively simple classical OFT methods also predict longer exploitation times.
We also show how explicitly including large initial in-patch costs in classical
OFT methods explains why sunk-cost effects like those observed by Nolet et al.
(2001) are rational rate-maximizing foraging behaviors. This result seems counterintuitive, but it follows from using a farsighted decision-making model based on
comparing present returns to the opportunity cost of not searching. When encounters
with costly patches are likely, it is better to spend more time in each patch in order
to reduce the accumulation rate of said costs. This interpretation may also explain
human preference for high price (e.g., Arkes and Blumer 1985) and the escalation of
costly behaviors ad infinitum (e.g., Dawkins and Carlisle 1976; Faver and Strand
2003). Sunk-cost behaviors are criticized because they are examples of making
decisions based on past investments; however, they should be better understood as
decisions that reduce the frequency of similar future investments.
Acknowledgments We thank Thomas A. Waite for his helpful insights and instruction and Ian
M. Hamilton for his comments on this paper. We also thank two anonymous referees for their help in
improving this paper. Additionally, the comments of three anonymous reviewers on a related submission
have also been influential in the presentation of this work.
References
Arkes HR, Ayton P (1999) The sunk cost and Concorde effects: are humans less rational than lower
animals? Psychol Bull 125(5):591–600
Arkes H, Blumer C (1985) The psychology of sunk cost. Organ Behav Hum Decis 35:124–140
Charnov EL (1973) Optimal foraging: some theoretical explorations. PhD thesis, University of
Washington
Charnov EL (1976) Optimal foraging: the marginal value theorem. Theor Popul Biol 9(2):129–136
Dawkins R, Carlisle TR (1976) Parental investment, mate desertion and a fallacy. Nature 262(5564):
131–133
Faver CA, Strand EB (2003) To leave or to stay? J Interpers Violence 18(12):1367–1377. doi:10.1177/
0886260503258028
Giraldeau LA, Caraco T (2000) Social foraging theory. Princeton University Press, Princeton
Giraldeau LA, Livoreil B (1998) Game theory and social foraging. In: Dugatkin LA, Reeve HK (eds)
Game theory and animal behavior. Oxford University Press, New York, pp 16–37
Houston AI, McNamara JM (1999) Models of adaptive behavior. Cambridge University Press, Cambridge
Jakob EM (2004) Individual decisions and group dynamics: why pholcid spiders join and leave groups.
Anim Behav 68(1):9–20 doi:10.1016/j.anbehav.2003.06.026
Kanodia C, Bushman R, Dickhaut J (1989) Escalation errors and the sunk cost effect: an explanation
based on reputation and information asymmetries. J Account Res 27(1):59–77
Nolet BA, Langevoord O, Bevan RM, Engelaar KR, Klaassen M, Mulder RJW, Dijk SV (2001) Spatial
variation in tuber depletion by swans explained by differences in net intake rates. Ecology
82(6):1655–1667 doi:10.1890/0012-9658(2001)082[1655:SVITDB]2.0.CO;2
Nonacs P (2001) State dependent behavior and the marginal value theorem. Behav Ecol 12(1):71–83
Olsson O, Brown JS (2006) The foraging benefits of information and the penalty of ignorance. Oikos
112(2):260–273 doi:10.1111/j.0030-1299.2006.13548.x
Olsson O, Holmgren NMA (1998) The survial-rate-maximizing policy for bayesian foragers: wait for
good news. Behav Ecol 9(4):345–353
123
66
T. P. Pavlic, K. M. Passino
Pavlic TP (2007) Optimal foraging theory revisited. Master’s thesis, The Ohio State University,
Columbus. http://www.ohiolink.edu/etd/view.cgi?acc_num=osu1181936683
Pyke GH, Pulliam HR, Charnov EL (1977) Optimal foraging: a selective review of theory and tests.
Q Rev Biol 52(2):137–154
Schoener TW (1971) Theory of feeding strategies. Annu Rev Ecol Syst 2:369–404
Sih A, Christensen B (2001) Optimal diet theory: when does it work, and when and why does it fail?
Anim Behav 61(2):379–390 doi:10.1006/anbe.2000.1592
Staw BM (1981) The escalation of commitment to a course of action. Acad Manag Rev 6(4):577–587
Stephens DW, Krebs JR (1986) Foraging theory. Princeton University Press, Princeton
van Gils JA, Schenk IW, Bos O, Piersma T (2003) Incompletely informed shorebirds that face a digestive
constraint maximize net energy gain when exploiting patches. Am Nat 161(5):777–793. doi:10.1086/
374205
van Gils JA, de Rooij SR, van Belle J, van der Meer J, Dekinga A, Piersma T, Drent R (2005) Digestive
bottleneck affects foraging decisions in red knots shape Calidris canutus. I. prey choice. J Anim
Ecol 74(1):105–119. doi:10.1111/j.1365-2656.2004.00903.x
123